On Sturm-Liouville Equations with Several Spectral Parameters

TL;DR

This paper derives explicit power series solutions for Sturm-Liouville equations with multiple spectral parameters, enabling numerical solutions and applications in optics.

Contribution

It generalizes known formulas for single spectral parameter cases to multiple parameters, providing recursive methods for solution coefficients.

Findings

Explicit formulas for solutions with multiple spectral parameters.

Numerical methods demonstrated for boundary value problems.

Application to optical transmission and reflectance analysis.

Abstract

We give explicit formulas for a pair of linearly independent solutions of $(py')'(x)+q(x)=(\lambda_1r_1(x)+\cdots+\lambda_dr_d(x))y(x)$, thus generalizing to arbitrary $d$ previously known formulas for $d=1$. These are power series in the spectral parameters $\lambda_1,\dots,\lambda_d$ (real or complex), with coefficients which are functions on the interval of definition of the differential equation. The coefficients are obtained recursively using indefinite integrals involving the coefficients of lower degree. Examples are provided in which these formulas are used to solve numerically some boundary value problems for $d=2$, as well as an application to transmission and reflectance in optics.